# NAME

Math::PlanePath::Staircase -- integer points in stair-step diagonal stripes

# SYNOPSIS

`````` use Math::PlanePath::Staircase;
my \$path = Math::PlanePath::Staircase->new;
my (\$x, \$y) = \$path->n_to_xy (123);``````

# DESCRIPTION

This path makes a staircase pattern down from the Y axis to the X,

``````     8      29
|
7      30---31
|
6      16   32---33
|         |
5      17---18   34---35
|         |
4       7   19---20   36---37
|         |         |
3       8--- 9   21---22   38---39
|         |         |
2       2   10---11   23---24   40...
|         |         |
1       3--- 4   12---13   25---26
|         |         |
Y=0 ->   1    5--- 6   14---15   27---28

^
X=0   1    2    3    4    5    6``````

The 1,6,15,28,etc along the X axis at the end of each run are the hexagonal numbers k*(2*k-1). The diagonal 3,10,21,36,etc up from X=0,Y=1 is the second hexagonal numbers k*(2*k+1), formed by extending the hexagonal numbers to negative k. The two together are the triangular numbers k*(k+1)/2.

Legendre's prime generating polynomial 2*k^2+29 bounces around for some low values then makes a steep diagonal upwards from X=19,Y=1, at a slope 3 up for 1 across, but only 2 of each 3 drawn.

## N Start

The default is to number points starting N=1 as shown above. An optional `n_start` can give a different start, in the same pattern. For example to start at 0,

``````    n_start => 0

28
29 30
15 31 32
16 17 33 34
6 18 19 35 36
7  8 20 21 37 38
1  9 10 22 23 ....
2  3 11 12 24 25
0  4  5 13 14 26 27``````

# FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

`\$path = Math::PlanePath::Staircase->new ()`
`\$path = Math::PlanePath::AztecDiamondRings->new (n_start => \$n)`

Create and return a new staircase path object.

`\$n = \$path->xy_to_n (\$x,\$y)`

Return the point number for coordinates `\$x,\$y`. `\$x` and `\$y` are rounded to the nearest integers, which has the effect of treating each point `\$n` as a square of side 1, so the quadrant x>=-0.5, y>=-0.5 is covered.

`(\$n_lo, \$n_hi) = \$path->rect_to_n_range (\$x1,\$y1, \$x2,\$y2)`

The returned range is exact, meaning `\$n_lo` and `\$n_hi` are the smallest and biggest in the rectangle.

# FORMULAS

## Rectangle to N Range

Within each row increasing X is increasing N, and in each column increasing Y is increasing pairs of N. Thus for `rect_to_n_range()` the lower left corner vertical pair is the minimum N and the upper right vertical pair is the maximum N.

A given X,Y is the larger of a vertical pair when ((X^Y)&1)==1. If that happens at the lower left corner then it's X,Y+1 which is the smaller N, as long as Y+1 is in the rectangle. Conversely at the top right if ((X^Y)&1)==0 then it's X,Y-1 which is the bigger N, again as long as Y-1 is in the rectangle.

# OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

``````    n_start=1 (the default)
A084849    N on diagonal X=Y
A210521    permutation N by diagonals, upwards
A199855      inverse

n_start=0
A014105    N on diagonal X=Y, second hexagonal numbers

n_start=2
A128918    N on X axis, except initial 1,1
A096376    N on diagonal X=Y``````

http://user42.tuxfamily.org/math-planepath/index.html